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$( Omega,F, P )$: a measurable space equiped with a finite measure

$(B , Vert cdot Vert) $ : a Banach space with $mathcalB$ as its borelian $sigma$-algebra

$p$ : a constant bigger than $1$

Define $L^p(Omega, B )$ the vector space that contain all $( F, mathcalB)$-measurable function $f$ such that :

$ vert Vert f Vert vert = sqrt[p] int_Omega Vert f Vert ^p cdot dP < infty$

I'm looking for a version of Riesz-Fischer theorem which affirms that:

Proposition:
$left( L^p(Omega, B ) , vert Vert cdot Vert vert right)$ is a Banach space

With some quick calculations, I have the feeling that this proposition is quite easy to be proved. But as we all know, it's always better to have a reliable reference.

So my question is: " Is the above proposition true? And does anyone have references to this matter?"

These are called Bochner spaces, and yes, they are Banach spaces.

At least one of the standard proofs that $L^p$ is complete goes through basically without change:

Let $f_n$ be Cauchy in this norm. Pass to a subsequence so that $||f_n - f_n+1|| le 4^-n$. By Chebyshev's inequality, we then have $P(|f_n - f_n+1| ge 2^-n) le 2^-pn$. Then the Borel-Cantelli lemma implies that for almost every $omega in Omega$, we have $|f_n(omega) - f_n+1(omega)| le 2^-n$ for all but finitely many $n$. In particular, for such $omega$, the sequence $f_n(omega)$ is Cauchy in $B$, so it converges to some $f(omega) in B$.

Now you have that $f$ is the a.e. limit of the $f_n$. Let $epsilon > 0$. Since $f_n$ is Cauchy in $||cdot||$-norm, choose $N$ so large that $||f_n - f_m|| < epsilon$ for all $n,m > N$. Letting $m to infty$ and using Fatou's lemma on the integrals $int_Omega |f_n - f_m|,dP$, conclude that $||f_n - f|| < epsilon$ as well. Thus the subsequence $f_n$ converges to $f$ in norm. Now use the Cauchy assumption one more time to see that the original sequence converges to $f$ as well.

I think that Evans's PDE book has some basic results about these spaces. There should be lots of other functional analysis texts that discuss them in more detail.

A beautiful treatment of vector valued $L^p$ spaces is in the book:

J. Diestel, J. J. Uhl, Vector measures. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977.